Definably simple groups in o-minimal structures
Y.
Peterzil;
A.
Pillay;
S.
Starchenko
4397-4419
Abstract: Let $\mathbb{G} =\langle G, \cdot\rangle$ be a group definable in an o-minimal structure $\mathcal{M}$. A subset $H$ of $G$ is $\mathbb{G}$-definable if $H$ is definable in the structure $\langle G,\cdot\rangle$(while definable means definable in the structure $\mathcal{M}$). Assume $\mathbb{G}$ has no $\mathbb{G}$-definable proper subgroup of finite index. In this paper we prove that if $\mathbb{G}$has no nontrivial abelian normal subgroup, then $\mathbb{G}$ is the direct product of $\mathbb{G}$-definable subgroups $H_1,\ldots,H_k$ such that each $H_i$ is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin's conjecture.
Simple algebraic and semialgebraic groups over real closed fields
Ya'acov
Peterzil;
Anand
Pillay;
Sergei
Starchenko
4421-4450
Abstract: We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In Definably simple groups in o-minimal structures, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.
Manifolds with minimal radial curvature bounded from below and big volume
Valery
Marenich
4451-4468
Abstract: We prove that a convergence in the Gromov-Hausdorff distance of manifolds with minimal radial curvature bounded from below by 1 to the standard sphere is equivalent to a volume convergence.
Willmore two-spheres in the four-sphere
Sebastián
Montiel
4469-4486
Abstract: Genus zero Willmore surfaces immersed in the three-sphere $\mathbb{S}^3$correspond via the stereographic projection to minimal surfaces in Euclidean three-space with finite total curvature and embedded planar ends. The critical values of the Willmore functional are $4\pi k$, where $k\in\mathbb{N}^*$, with $k\ne 2,3,5,7$. When the ambient space is the four-sphere $\mathbb{S}^4$, the regular homotopy class of immersions of the two-sphere $\mathbb{S}^2$ is determined by the self-intersection number $q\in\mathbb{Z}$; here we shall prove that the possible critical values are $4\pi (\vert q\vert+k+1)$, where $k\in\mathbb{N}$. Moreover, if $k=0$, the corresponding immersion, or its antipodal, is obtained, via the twistor Penrose fibration $\mathbb{P}^3\rightarrow \mathbb{S}^4$, from a rational curve in $\mathbb{P}^3$and, if $k\ne 0$, via stereographic projection, from a minimal surface in $\mathbb{R}^4$ with finite total curvature and embedded planar ends. An immersion lies in both families when the rational curve is contained in some $\mathbb{P}^2\subset\mathbb{P}^3$ or (equivalently) when the minimal surface of $\mathbb{R}^4$ is complex with respect to a suitable complex structure of $\mathbb{R}^4$.
Principal curvatures of isoparametric hypersurfaces in $\mathbb{C}P^{n}$
Liang
Xiao
4487-4499
Abstract: Let $M$ be an isoparametric hypersurface in $\mathbb{C}P^{n}$, and $\overline{M}$ the inverse image of $M$ under the Hopf map. By using the relationship between the eigenvalues of the shape operators of $M$ and $\overline{M}$, we prove that $M$ is homogeneous if and only if either $g$or $l$ is constant, where $g$ is the number of distinct principal curvatures of $M$ and $l$ is the number of non-horizontal eigenspaces of the shape operator on $\overline{M}$.
Symplectic 4-manifolds with Hermitian Weyl tensor
Vestislav
Apostolov;
John
Armstrong
4501-4513
Abstract: It is proved that any compact almost Kähler, Einstein 4-manifold whose fundamental form is a root of the Weyl tensor is necessarily Kähler.
The toric $h$-vectors of partially ordered sets
Margaret
M.
Bayer;
Richard
Ehrenborg
4515-4531
Abstract: An explicit formula for the toric $h$-vector of an Eulerian poset in terms of the $\mathbf{cd}$-index is developed using coalgebra techniques. The same techniques produce a formula in terms of the flag $h$-vector. For this, another proof based on Fine's algorithm and lattice-path counts is given. As a consequence, it is shown that the Kalai relation on dual posets, $g_{n/2}(P)=g_{n/2}(P^*)$, is the only equation relating the $h$-vectors of posets and their duals. A result on the $h$-vectors of oriented matroids is given. A simple formula for the $\mathbf{cd}$-index in terms of the flag $h$-vector is derived.
Examples of torsion points on genus two curves
John
Boxall;
David
Grant
4533-4555
Abstract: We describe a method that sometimes determines all the torsion points lying on a curve of genus two defined over a number field and embedded in its Jacobian using a Weierstrass point as base point. We then apply this to the examples $y^{2}=x^{5}+x$, $y^{2}=x^{5}+5\,x^{3}+x$, and $y^{2}-y=x^{5}$.
$\mathbf{C}^{2}$-saddle method and Beukers' integral
Masayoshi
Hata
4557-4583
Abstract: We give good non-quadraticity measures for the values of logarithm at specific rational points by modifying Beukers' double integral. The two-dimensional version of the saddle method, which we call $\mathbf{C}^{2}$-saddle method, is applied.
The Mod-2 cohomology of the Bianchi groups
Ethan
Berkove
4585-4602
Abstract: The Bianchi groups are a family of discrete subgroups of $PSL_2(\mathbb C)$which have group theoretic descriptions as amalgamated products and HNN extensions. Using Bass-Serre theory, we show how the cohomology of these two constructions relates to the cohomology of their pieces. We then apply these results to calculate the mod-2 cohomology ring for various Bianchi groups.
A global approach to fully nonlinear parabolic problems
Athanassios
G.
Kartsatos;
Igor
V.
Skrypnik
4603-4640
Abstract: We consider the general initial-boundary value problem (1) $\displaystyle{\frac{\partial u}{\partial t}-F(x,t,u,\mathcal{D}^{1}u, \mathcal{D}^{2}u)=f(x,t),\quad (x,t)\in Q_{T}\equiv \Omega \times (0,T),}$ (2) $\displaystyle{G(x,t,u,\mathcal{D}^{1}u)=g(x,t),\quad (x,t)\in S_{T}\equiv \partial\Omega \times (0,T),}$ (3) $\displaystyle{u(x,0)=h(x),\quad x\in \Omega,}$ where $\Omega$ is a bounded open set in $\mathcal{R}^{n}$ with sufficiently smooth boundary. The problem (1)-(3) is first reduced to the analogous problem in the space $W^{(4),0}_{p}(Q_{T})$with zero initial condition and \begin{displaymath}f\in W^{(2),0}_{p}(Q_{T}),~g \in W^{(3-\frac{1}{p}),0}_{p}(S_{T}). \end{displaymath} The resulting problem is then reduced to the problem $Au=0,$ where the operator $A:W^{(4),0}_{p}(Q_{T})\to \left [W^{(4),0}_{p}(Q_{T})\right ]^{*}$ satisfies Condition $(S)_{+}.$ This reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in Sobolev spaces. The local and global solvability of the operator equation $Au=0$ are achieved via topological methods developed by I. V. Skrypnik. Further applications are also given involving relevant coercive problems, as well as Galerkin approximations.
Invariant foliations near normally hyperbolic invariant manifolds for semiflows
Peter
W.
Bates;
Kening
Lu;
Chongchun
Zeng
4641-4676
Abstract: Let $M$ be a compact $C^1$ manifold which is invariant and normally hyperbolic with respect to a $C^1$ semiflow in a Banach space. Then in an $\epsilon$-neighborhood of $M$ there exist local $C^1$ center-stable and center-unstable manifolds $W^{cs}(\epsilon)$ and $W^{cu}(\epsilon)$, respectively. Here we show that $W^{cs}(\epsilon)$ and $W^{cu}(\epsilon)$ may each be decomposed into the disjoint union of $C^1$ submanifolds (leaves) in such a way that the semiflow takes leaves into leaves of the same collection. Furthermore, each leaf intersects $M$ in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time.
Polynomials that are positive on an interval
Victoria
Powers;
Bruce
Reznick
4677-4692
Abstract: This paper discusses representations of polynomials that are positive on intervals of the real line. An elementary and constructive proof of the following is given: If $h(x), p(x) \in \mathbb{R}[x]$ such that $\{ \alpha \in \mathbb{R} \mid h(\alpha) \geq 0 \} = [-1,1]$ and $p(x) > 0$ on $[-1,1]$, then there exist sums of squares $s(x), t(x) \in \mathbb{R}[x]$ such that $p(x) = s(x) + t(x) h(x)$. Explicit degree bounds for $s$ and $t$ are given, in terms of the degrees of $p$ and $h$ and the location of the roots of $p$. This is a special case of Schmüdgen's Theorem, and extends classical results on representations of polynomials positive on a compact interval. Polynomials positive on the non-compact interval $[0,\infty)$ are also considered.
The optimal differentiation basis and liftings of $L^{{\infty}}$
Jürgen
Bliedtner;
Peter
A.
Loeb
4693-4710
Abstract: There is an optimal way to differentiate measures when given a consistent choice of where zero limits must occur. The appropriate differentiation basis is formed following the pattern of an earlier construction by the authors of an optimal approach system for producing boundary limits in potential theory. Applications include the existence of Lebesgue points, approximate continuity, and liftings for the space of bounded measurable functions - all aspects of the fact that for every point outside a set of measure $0$, a given integrable function has small variation on a set that is ``big'' near the point. This fact is illuminated here by the replacement of each measurable set with the collection of points where the set is ``big'', using a classical base operator. Properties of such operators and of the topologies they generate, e.g., the density and fine topologies, are recalled and extended along the way. Topological considerations are simplified using an extension of base operators from algebras of sets on which they are initially defined to the full power set of the underlying space.
Periodic points of holomorphic maps via Lefschetz numbers
Núria
Fagella;
Jaume
Llibre
4711-4730
Abstract: In this paper we study the set of periods of holomorphic maps on compact manifolds, using the periodic Lefschetz numbers introduced by Dold and Llibre, which can be computed from the homology class of the map. We show that these numbers contain information about the existence of periodic points of a given period; and, if we assume the map to be transversal, then they give us the exact number of such periodic orbits. We apply this result to the complex projective space of dimension $n$ and to some special type of Hopf surfaces, partially characterizing their set of periods. In the first case we also show that any holomorphic map of ${\mathbb CP}(n)$ of degree greater than one has infinitely many distinct periodic orbits, hence generalizing a theorem of Fornaess and Sibony. We then characterize the set of periods of a holomorphic map on the Riemann sphere, hence giving an alternative proof of Baker's theorem.
Endomorphisms of expansive systems on compact metric spaces and the pseudo-orbit tracing property
Masakazu
Nasu
4731-4757
Abstract: We investigate the interrelationships between the dynamical properties of commuting continuous maps of a compact metric space. Let $X$ be a compact metric space. First we show the following. If $\tau: X \rightarrow X$ is an expansive onto continuous map with the pseudo-orbit tracing property (POTP) and if there is a topologically mixing continuous map $\varphi: X \rightarrow X$ with $\tau\varphi = \varphi\tau$, then $\tau$ is topologically mixing. If $\tau: X \rightarrow X$ and $\varphi: X \rightarrow X$ are commuting expansive onto continuous maps with POTP and if $\tau$ is topologically transitive with period $p$, then for some $k$dividing $p$, $X = \bigcup_{i=0}^{l-1} B_i$, where the $B_i$, $0 \leq i \leq l-1$, are the basic sets of $\varphi$ with $l = p/k$ such that all $\varphi\vert B_i : B_i \rightarrow B_i$ have period $k$, and the dynamical systems $(B_i,\varphi\vert B_i)$ are a factor of each other, and in particular they are conjugate if $\tau$ is a homeomorphism. Then we prove an extension of a basic result in symbolic dynamics. Using this and many techniques in symbolic dynamics, we prove the following. If $\tau: X \rightarrow X$ is a topologically transitive, positively expansive onto continuous map having POTP, and $\varphi: X \rightarrow X$ is a positively expansive onto continuous map with $\varphi\tau = \tau\varphi$, then $\varphi$ has POTP. If $\tau:X \rightarrow X$ is a topologically transitive, expansive homeomorphism having POTP, and $\varphi : X \rightarrow X$ is a positively expansive onto continuous map with $\varphi\tau = \tau\varphi$, then $\varphi$ has POTP and is constant-to-one. Further we define `essentially LR endomorphisms' for systems of expansive onto continuous maps of compact metric spaces, and prove that if $\tau: X \rightarrow X$ is an expansive homeomorphism with canonical coordinates and $\varphi$ is an essentially LR automorphism of $(X,\tau)$, then $\varphi$ has canonical coordinates. We add some discussions on basic properties of the essentially LR endomorphisms.
An equivariant Brauer semigroup and the symmetric imprimitivity theorem
Astrid
an Huef;
Iain
Raeburn;
Dana
P.
Williams
4759-4787
Abstract: Suppose that $(X,G)$ is a second countable locally compact transformation group. We let $\operatorname{S}_G(X)$ denote the set of Morita equivalence classes of separable dynamical systems $(A,G,\alpha)$ where $A$ is a $C_{0}(X)$-algebra and $\alpha$ is compatible with the given $G$-action on $X$. We prove that $\operatorname{S}_{G}(X)$ is a commutative semigroup with identity with respect to the binary operation $[A,G,\alpha][B,G,\beta]=[A\otimes_{X}B,G,\alpha\otimes_{X}\beta]$ for an appropriately defined balanced tensor product on $C_{0}(X)$-algebras. If $G$and $H$ act freely and properly on the left and right of a space $X$, then we prove that $\operatorname{S}_{G}(X/H)$ and $\operatorname{S}_{H}(G\backslash X)$ are isomorphic as semigroups. If the isomorphism maps the class of $(A,G,\alpha)$to the class of $(B,H,\beta)$, then $A\rtimes_{\alpha}G$ is Morita equivalent to $B\rtimes_{\beta}H$.
$q$-Krawtchouk polynomials as spherical functions on the Hecke algebra of type $B$
H.
T.
Koelink
4789-4813
Abstract: The Hecke algebra for the hyperoctahedral group contains the Hecke algebra for the symmetric group as a subalgebra. Inducing the index representation of the subalgebra gives a Hecke algebra module, which splits multiplicity free. The corresponding zonal spherical functions are calculated in terms of $q$-Krawtchouk polynomials using the quantised enveloping algebra for ${\mathfrak{sl}}(2,\mathbb{C} )$. The result covers a number of previously established interpretations of ($q$-)Krawtchouk polynomials on the hyperoctahedral group, finite groups of Lie type, hypergroups and the quantum $SU(2)$ group.
Local structure of Schelter-Procesi smooth orders
Lieven
Le Bruyn
4815-4841
Abstract: In this paper we give the étale local classification of Schelter-Procesi smooth orders in central simple algebras. In particular, we prove that if $\Delta$ is a central simple $K$-algebra of dimension $n^2$, where $K$is a field of trancendence degree $d$, then there are only finitely many étale local classes of smooth orders in $\Delta$. This result is a non-commutative generalization of the fact that a smooth variety is analytically a manifold, and so has only one type of étale local behaviour.
A reduced Tits quadratic form and tameness of three-partite subamalgams of tiled orders
Daniel
Simson
4843-4875
Abstract: Let $D$ be a complete discrete valuation domain with the unique maximal ideal ${\mathfrak{p}}$. We suppose that $D$ is an algebra over an algebraically closed field $K$ and $D/{\mathfrak{p}} \cong K$. Subamalgam $D$-suborders $\Lambda ^{\bullet }$ of a tiled $D$-order $\Lambda$ are studied in the paper by means of the integral Tits quadratic form $q_{\Lambda ^{\bullet }}: {\mathbb{Z} }^{n_{1}+2n_{3}+2 } \,\,\longrightarrow {\mathbb{Z} }$. A criterion for a subamalgam $D$-order $\Lambda ^{\bullet }$ to be of tame lattice type is given in terms of the Tits quadratic form $q_{{\Lambda ^{\bullet }}}$ and a forbidden list $\Omega _{1},\ldots ,\Omega _{17}$ of minor $D$-suborders of $\Lambda ^{\bullet }$presented in the tables.
Skein modules and the noncommutative torus
Charles
Frohman;
Razvan
Gelca
4877-4888
Abstract: We prove that the Kauffman bracket skein algebra of the cylinder over a torus is a canonical subalgebra of the noncommutative torus. The proof is based on Chebyshev polynomials. As an application, we describe the structure of the Kauffman bracket skein module of a solid torus as a module over the algebra of the cylinder over a torus, and recover a result of Hoste and Przytycki about the skein module of a lens space. We establish simple formulas for Jones-Wenzl idempotents in the skein algebra of a cylinder over a torus, and give a straightforward computation of the $n$-th colored Kauffman bracket of a torus knot, evaluated in the plane or in an annulus.